Chi-square (χ 2 ) Tests

Transcription

1 Math Mathematical Statistics II April 30, 2018 Chi-square (χ 2 ) Tests Common Uses of the χ 2 test. 1. Testing Goodness-of-fit. 2. Testing Equality of Several Proportions. 3. Homogeneity Test. 4. Testing Independence. Testing Goodness-of-fit of Data from Multinomial Experiment. Properties of the Multinomial Experiment. 1. The experiment consists of n identical trials. 2. There are k possible outcomes to each trial. These possible outcomes are sometimes called classes or categories. 3. The probabilities of the k outcomes, denoted by p 1, p 2,..., p k, remain the same from trial to trial, and p 1 + p p k The trials are independent. 5. The random variables of interest are the cell counts, n 1, n 2,..., n k, of the number of observations that fall in each of the k categories. (note: n 1 + n n k n). Cat 1 Cat 2 Cat k Observed n 1 n 2 n k Expected n p 1 n p 2 n p k Chi-Square Statistic. The Chi-square statistic is a measure of how much the observed cell counts diverge from the expected cell counts. In this case, we use the one-dimensional χ 2 statistic because only one factor is being investigated. The formula for this statistic is X 2 k (observed count expected count) 2 expected count k (O E) 2 E i1 i1 k (n i n p i ) 2 i1 (1) (2) n p i χ 2 (k 1). (3) This X 2 statistic follows approximately the χ 2 distribution with k 1 degrees of freedom. Conditions Required for χ 2 Test to be valid: 1. A multinomial experiment has been conducted. This is generally satisfied by taking a random sample from the population of interest. 2. The sample size n is large enough so that the expected count for each cell is at least 5. Sometimes two or more categories are collapsed together to form a bigger category with higher expected count. Null Hypothesis and Alternative Hypothesis. 1. Null Hypothesis: H o : p 1 p o1, p 2 p o2,..., p k p ok [Note: p o1 + p o2 + + p ok 1] 2. Alternative Hypothesis: H 1 : The null hypothesis is not true.

2 Examples: 1. Nature (Sept. 1993) reported on a study of animal and plant species hotspots in Great Britain. A hotspot is defined as a 10-km 2 area that is species-rich, that is, is heavily populated by the species of interest. Analogously, a coldspot is a 10-km 2 area that is species-poor. The table below gives the number of butterfly hotspots and the number of butterfly coldspots in a sample of 2, km 2 areas. In theory, 5% should be butterfly hotspots and 5% coldspots, while the remaining areas (90%) are neutral. Test the theory using α Hotspots Coldspots Neutral Areas Total Observed (Expected) 123 ( ) 147 ( ) 2318 ( ) Car Crashes and Age Brackets. Among drivers who have had a car crash in the last year, 88 are randomly selected and categorized by age, with the results listed in the accompanying table. If all ages have the same crash rate, we would expect (because of the age distribution of licensed drivers) the given categories to have 16%, 44%, 27%, and 13% of the subjects, respectively. At the 0.05 level of significance, test the claim that the distribution of crashes conforms to the distribution of ages. Does any age group appear to have a disproportionate number of crashes? Age Under Over 64 No. of Crashes 36 ( ) 21 ( ) 12 ( ) 19 ( ) 3. Checking Normality. The table below shows the number of values in a sample of size n 200 observations falling in each category. At the 0.05 level of significance, test whether the sample can be reasonably assumed to have come from the Standard Normal distribution. z values z < 1 1 z <.5.5 z < 0 0 z <.5.5 z < 1 z > 1 No. of Obs

3 Testing Equality of Several Proportions. 1. Null Hypothesis: H 0 : p 1 p 2 p k 2. Alternative Hypothesis: H 1 : Not all are equal. 3. Test Statistic: The Two-Factor Chi-square Statistic (X 2 ) for Two-Way (r c) Contingency Table. The formula for the statistic is very similar to the one we have before and is given below: X 2 (observed count expected count) 2 expected count (4) r c (O ij E ij ) 2 r c (n ij r i ˆp j ) 2 E ij r i ˆp j (5) all cells i1 j1 r c i1 j1 (n ij (r i c j )/n) 2 (r i c j )/n i1 j1 χ 2 (r 1)(c 1). (6) where, r i and c j are the total of the ith row and jth column, respectively. This X 2 statistic follows approximately the χ 2 distribution with (r 1)(c 1) degrees of freedom. 4. Conditions Required for χ 2 Test to be valid: Examples. a. The n observed counts are a random sample from the population of interest. b. The sample size n is large enough so that the expected count for each cell is at least 5. Sometimes two or more categories are collapsed together to form a bigger category with higher expected count. 1. Quitting Smoking. The accompanying table summarizes successes and failures when subjects used different methods in trying to stop smoking. The determination of smoking or not smoking was made five months after the treatment was begun, and the data are based on results from the Centers for Disease Control and Prevention. Use a 0.05 level of significance to test the claim that success is independent of the method used. If someone wants to stop smoking, does the choice of the method make a difference? Nicotine Gum Nicotine Patch Nicotine Inhaler Smoking No smoking

4 2. Survey Refusals and Age Bracket. A study of people who refused to answer survey questions provided the randomly selected sample data shown in the table. At the 0.01 significance level, test the claim that the cooperation of the subject (response or refusal) is independent of the age category. Does any particular age group appear to be particularly uncooperative? Age and over Responded Refused Testing for Homogeneity (In a Two-Way Contingency Tables). 1. Null Hypothesis: H 0 : p 1j p 2j p kj, j 1, 2,..., J 2. Alternative Hypothesis: H 1 : H 0 is not true. 3. Test Statistic: The chi-square statistic X 2. Example. 1. A company packages a particular product in cans of three different sizes, each one using a different production line. Most cans conform to specifications, but a quality control engineer has identified the following reasons for non-conformance: Blemish on can, crack in can, improper pull tab location, pull tab missing, and other. A sample of non-conforming units is selected from each of the three lines, and each unit is categorized according to reason for nonconformity, resulting in the contingency table below. Using α 0.05, test the hypothesis that the distributions of non-conformance of the three production lines are homogeneous. Production Blemish Crack Location Missing Other Total Line Line Line Total

5 Testing for Independence (In a Two-Way Contingency Tables). 1. Null Hypothesis: Factor A is independent of factor B. 2. Alternative Hypothesis: Factors A and B are not independent. 3. Test Statistic: The same chi-square statistic X 2. Example. 1. Background Music. Market researchers know that background music can influence the mood and purchasing behavior of customers. One study in a supermarket in Northern Ireland compared three treatments: no music, French accordion music, and Italian string music. Under each condition, the researchers recorded the numbers of bottles of French, Italian, and other wine purchased. The twoway table that summarizes the data are given below. Test the hypothesis that background music and purchasing behavior (type of wine purchased) are independent. Use α Wine French music Italian music No music French Italian Other